Perron’s capacity of random sets

Abstract

We answer in a probabilistic setting two questions raised by Stokolos in a private communication. Precisely, given a sequence of random variables $\left\{X_k : k \geq 1\right\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions \begin{equation*}\Omega_{\text{rand},\text{lin}} := \left\{ \frac{\pi X_k}{k}: k \geq 1\right\}\end{equation*} and \begin{equation*}\Omega_{\text{rand},\text{lac}} := \left\{\frac{\pi X_k}{2^k} : k\geq 1 \right\}.\end{equation*} We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p({\mathbb{R}}^2)$ for any $1 \lt p \lt \infty$ .